Question

If $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$ represents a equation of circle, then the value of $\lambda$ when the radius of circle is minimum, is

Answer 1

Given : $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$
Let $z=x+iy$, then
$(x-2{)}^2+(y-3{)}^2+(x-4{)}^2+(y-3{)}^2=\lambda \Rightarrow 2x^2+2y^2-12x-12y=\lambda -38\Rightarrow x^2+y^2-6x-6y-\frac{\lambda -38}{2}=0$
The minimum possible radius of the circle is $0$, so
$\sqrt{9+9+\frac{\lambda -38}{2}}=0\therefore \lambda =2$